
Re: Fundamental period for complex functions
Posted:
Aug 12, 2013 11:00 AM


steinerartur@gmail.com wrote: > > Suppose f from the complexes to the complexes is periodic. Then, is > there something like a fundamental period for f? > > On the reals, if f is periodic then f may have a fundamental period, > defined, when it exists, as the smallest period of f. By definition, > periods are positive, and, if P is the set of periods of f, then the > fundamental period is p*= infimum P, if p* > 0. In this case, we can > show p* is in P. So, in this case, p* is the minimum of P. And it's > known that if f is continuous, periodic and non constant, then f has > a fundamental period (aka minimum period). > > In the complexes, the above definition doesn't make sense, but maybe > we can define p* as the period with minimum absolute value, provided > it's positive. So, the definition might be p = minimum {p : p is > period of f}. Does this exist? > > For example, 2?i is a period of f(z) = e^z. Does f have a period with > positive absolute value < 2?? > > If we define p* for the complexes in this way, then, supposing f is > continuous, periodic and non constant, then does such p* exist? > > Thank you
You may like to read https://en.wikipedia.org/wiki/Doubly_periodic_function and https://en.wikipedia.org/wiki/Fundamental_pair_of_periods.
 Nam Nguyen in sci.logic in the thread 'Q on incompleteness proof' on 16/07/2013 at 02:16: "there can be such a group where informally it's impossible to know the truth value of the abelian expression Axy[x + y = y + x]".

